Possibly shouldn't be an answer, but...
The TL/12 (original, 1947) has separate bias (cathode resistor) for each valve; the others generally used a common cathode resistor and there, you do need reasonably well matched valves. You are correct that none (that I know of) used negative grid bias supply, sorry if I misled you on that.
I'm puzzled how you expect the two halves of a classic push-pull amplifier would interact. Each has its own DC feed via half the transformer primary; its own cathode resistor (if automatic bias, except as above) or its own grid bias (if not).
For DC purposes, these are practically independent circuits. Interactions between them are pretty much limited to variation in the main supply voltage, at the centre tap of the primary - which introduces second order effects.
It's not like a transistor amplifier (excluding the transformer coupled ones!) where both halves of the push-pull pair are in series in the same circuit and cannot be biased independently.
So the procedure (for classic push-pull amplifiers) is :
bias first valve to required anode current
bias second valve likewise
check that first has not shifted appreciably
done
And for the Leaks, with automatic bias, that just meant check the voltages across the cathode resistor. Whether you simply replaced valves if out-of-spec or tried to get another year by fiddling the resistor was up to you and your bank account (or valued sources of parts!)
Mismatch between the anode currents will result in modest levels of even harmonic distortion. The two DC currents in the transformer balance out so there is no reduction in available flux before saturation, unless the mismatch is gross.
I've seen people (wearing gloves!) pull a valve out and plug in another without noticably distressing the amp (Not recommended though!) I even ran a TL/12 for a couple of poverty-stricken years with an EL34 and a KT66. Sounded lovely, but looked a bit too much like Laurel & Hardy!
Now there may be some reason why the Thorens gives unusual grief, but so far I can't see it. If this is the right schematic the bits I can see look like straightforward independent automatic bias (like the TL/12), and I'd guess the grids are grounded, giving about 35mA in each anode.
It's possible that the TV put all of the heaters in series directly across the AC line input. Very common in cheap "hot chassis" designs.
There's no reason you couldn't add a second power transformer yourself. 12.6 VAC transformers are very readily available.
Or you could use a separate DC 12V supply for testing purposes — Heaters don't care whether they're getting AC or DC.
Best Answer
The thermistor idea works well, I use it on all my amplifiers applied to the primary of the main power supply transformer. Connected this way it also limits the current inrush:
This said, it will only limit the current for 1 or 2 seconds, you may wish to rise the voltage slowly on tubes anodes. This is a bad idea to wait for the filaments to be hot and apply the B+ suddenly because:
The idea is to apply the HT gradually. Using a vacuum rectifier is a way to achieve this. You can also set up a mosfet regulated power supply:
Q1 and R1 create a CCS. This current will charge C1 through R2 and polarize Q2. When C1 is charged the voltage on the source of Q2 is roughly equivalent to Iccs * R
I use DN3545 for Q1 and IRF820 (up to 500V) for Q2. The zener is here to discharge C1 when the PSU is switched off and avoid Vgs > 20V (maximum specified by the DS).